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Image fusion and denoising using fractional-order gradient information

Mei, Jin-Jin; Dong, Yiqiu; Huang, Ting-Zhu

Publication date:2017

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Citation (APA):Mei, J-J., Dong, Y., & Huang, T-Z. (2017). Image fusion and denoising using fractional-order gradientinformation. Technical University of Denmark (DTU). (DTU Compute-Technical Report-2017, Vol. 05).

Image fusion and denoising using fractional-order gradientinformation ∗

Jin-Jin Mei† Yiqiu Dong‡ Ting-Zhu Huang§

Abstract

Image fusion and denoising are significant in image processing because of the availabilityof multi-sensor and the presence of the noise. The first-order and second-order gradient infor-mation have been effectively applied to deal with fusing the noiseless source images. In thispaper, due to the advantage of the fraction-order derivative, we first integrate the fractional-order gradients of noisy source images as the target fraction-order feature, and make it fitwith the fractional-order gradient of the fused image. Then we introduce the total variation(TV) regularization for removing the noise. By adding the data fitting term between the fusedimage and a preprocessed image, a new convex variational model is proposed for fusing thenoisy source images. Furthermore, an alternating direction method of multiplier (ADMM)is developed for solving the proposed variational model. Numerical experiments show thatthe proposed method outperforms the conventional total variation methods for simultaneouslyfusing and denoising.

Keywords: image fusion and denoising, alternating direction method of multiplier, inverseproblem, fractional-order derivative, structure tensor

1 Introduction

Image fusion is an important issue due to the availability of multi-sensor data in various fields suchas concealed weapon detection, remote sensing, medical diagnosis, defect inspection, and militarysurveillance [1–3]. Image fusion is to synthesize several source images of the same scene into asingle image which contains much more visual information. But, the observed source images areinevitably corrupted by the noise during the process of image collection, acquisition, transmissionand storage. For obtaining a high-quality image, it is necessary to simultaneously fuse and denoisethe source images.

Researchers have paid great attention to image fusion and studied various efficient algorithms.According to the different degree of information extracted from the source images, image fusionmethods are typically divided into three levels: pixel-level methods, feature-level methods anddecision-making level methods. The pixel-level methods are to directly estimate the pixel valuesof the fused image by integrating the pixel values of several source images with a feature selectionrule [4,5]. In this paper, since it preserves more original information from the source images [6,7],

∗This research was supported by 973 Program (2013CB329404), NSFC (61370147, 61402082, 11401081) and theFundamental Research Funds for the Central Universities (ZYGX2016J129).

†School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731,People’s Republic of China ( meijinjin666@126.com).

‡Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lyngby,Denmark (yido@dtu.dk). The work was supported by Advanced Grant 291405 from the European Research Council.

§School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731,People’s Republic of China (tingzhuhuang@126.com).

1

we devote to the study of the image fusion methods based on the pixel-level. Generally, the pixel-level fusion approaches mainly consist of spatial domain fusion methods [8–12] and transformdomain fusion methods [13–19]. Among the spatial domain fusion methods, some variationalmethods combined the structure tensor with gradient information have been studied for dealingwith the image fusion problem [11, 12, 20].

Before introducing the pixel-level fusion approaches, we firstly review some notations withrespect to the structure tensor. Suppose that Ω ⊂ R2 is a connected bounded image domainwith the compacted Lipschitz boundary, u : Ω → R is the unknown fused image, fi : Ω → R,i = 1, 2, · · · , l are the source images. According to [11, 21, 22], the structure tensor is used fordepicting the local structure feature, which is defined as follows

G(x, y) ,

∑l

i=1

(∂ fi∂x

)2 ∑li=1

∂ fi∂x

∂ fi∂y∑l

i=1∂ fi∂y

∂ fi∂x

∑li=1

(∂ fi∂y

)2

, ∀(x, y) ∈ Ω.

Then, the target gradient V : Ω → R2 consists of eigenvalues and eigenvectors of the structuretensor G. Since G is positive semi-definite, the target gradient V is formulated as

V ,√λe

where λ denotes the largest eigenvalues of G which conveys the sharp information, e is the corre-sponding unit eigenvector which indicates the orientation maximizing the pixel-value fluctuationsand satisfies that 〈e,

∑li=1 ∇ fi〉 ≥ 0. Based on the above target gradient of source images, a classic

variational model is proposed as

minu

∫Ω

|∇u − V |2dx. (1)

Note that the model (1) results in the decrease of the contrast and the solution is not unique.In [11], Piella combined the image enhancement techniques and proposed a variational model forintegrating the salient structure of source images and enhancing the contrast of the fused image.The author derived the corresponding Euler-Lagrange equation and applied the gradient descentmethod for obtaining the unknown fused image. But due to the slow convergent speed, the gradientdescent method is extremely time-consuming. In [23], the authors considered the first-order andsecond-order gradient information at different directions. By adding a fidelity term which requiresthat the fused image is close to a predefined image u0. They presented a new variational approach(called FSGF for short),

minu

∫Ω

(|∇u − sv| + α|∇2u − sw|

)dx +

β

2

∫Ω

|u − u0|2dx (2)

where α ≥ 0, β > 0, s ≥ 1, v and w denote the target gradients which integrated the first-order andsecond-order gradient information of the source images, respectively. A new feature selection wasused to construct the target gradient v and w. Note that if s = 1, the model (2) is just for imagefusion; if s > 1, the model is for simultaneous image fusion and enhancement. Moreover, the splitBregman method [24–26] was applied to solve the model (2).

As far as we know, a few researchers have paid attention to study the effective variationalapproaches for fusing the noisy source images. In [20], combining with the TV regularization,Wang et al. proposed a variational model for fusing and denoising the noisy multi-focus images(just referred to as WFTV),

minu

∫Ω

(α1(x)(u − f1)2 + α2(x)(u − f2)2

)dx + 2β

∫Ω

|∇u|dx (3)

2

where α1(x) and α2(x) are two nonnegative weight functions such that α1(x) + α2(x) = 1, f1 andf2 are the noisy source images, β is the positive parameter which trades off the fidelity term forfusing the noisy multi-focus images and the TV regularization for smoothing the fused image.They applied the gradient descent method to deal with (3). In [27], the authors proposed a adap-tive sparse representation model for image fusion and denoising. Assume that the high-qualityimage patches were classified into several categories in advance. Then, based on the gradient in-formation of images, they learned a set of compacted sub-dictionaries from numerous high-qualityimage patches and designed an algorithm to select adaptively one of the sub-dictionaries. Theirmethod was based on dictionary learning, not belong to the variational method. And it was fairlycomplicated and time consuming. However, in the following, we only focus on the variationalmethods for image fusion and denoising.

Recently, in [28], the authors illustrated the theoretical properties of the total fraction-ordervariation. Based on the the total fraction-order variation, their proposed model could preserve theedges and smoothness. Inspired by this advantage, we combine the fraction-order derivative andpropose a novel variational model for image fusion and denoising. According to [29–31], we buildthe target fractional-order gradient information from the noisy source images by a simple saliencecriterion [23]. For fusing the source images, we require that the fractional-order gradient informa-tion of the fused image should be matched with the target fractional-order gradient information inthe sense of L2 norm. Meanwhile, the fused image is closed to the predefined image. Since thesource images are corrupted by the noise, we add the TV regularization to the variational modelfor the regularity constrain. Furthermore, we prove the existence and uniqueness of the solution.Under the framework of ADMM [32–34], a fast algorithm is developed for solving the proposedmodel.

The rest of this letter is organized as follows. In the next section, we introduce the elemen-tary properties of the fractional-order derivative and the framework of ADMM. In Section 3, weobtain the target the fractional-order feature from the noisy source images and construct a newvariational model with the TV regularization for image fusion and denoising. Meanwhile we givethe existence and the uniqueness of the solution. In Section 4, we apply the ADMM algorithm tosolve the proposed model. Numerical experiments are shown in Section 5. Finally, we concludethe paper with a summary.

2 Preliminaries

In this section, for the completeness of the paper, we briefly review the fraction-order derivativeand the classic ADMM algorithm.

2.1 Fractional-order derivative

As a generalization of the integer-order derivatives, the fractional-order derivative has been widelyused in image processing [28, 35–37]. Firstly, the definition and property of the fractional-orderderivative have been studied over the one-dimensional domain, see [29–31] and references therein.Assume that a fraction γ ∈ R+ and a function h ∈ L1([a, b];R), the left and right Riemann-Liouville(RL) fractional-order derivatives are respectively defined by

aDγx[h](x) =

1Γ(1 − γ)

ddx

∫ x

a

h(τ)(x − τ)γ

dτ, x ∈ (a, b]

xDγb[h](x) =

−1Γ(1 − γ)

ddx

∫ b

x

h(τ)(τ − x)γ

dτ, x ∈ [a, b)

3

where Γ(γ) represents a Gamma function. Based on the left and the right Riemann-Liouvillefractional-order derivatives, the Riesz-RL (central) fractional-order derivative is given by

aDγb[h](x) =

12

(aDγ

x[h](x) − xDγb[h](x)

), x ∈ (a, b).

Similarly, the left, the right and the Riesz-Caputo fraction-order derivatives are defined by asfollows

Ca Dγ

x[h](x) =1

Γ(1 − γ)

∫ x

a

h′(τ)dτ(x − τ)γ

, x ∈ (a, b]

Cx Dγ

b[h](x) = −1

Γ(1 − γ)

∫ b

x

h′(τ)dτ(τ − x)γ

, x ∈ [a, b)

Ca Dγ

b[h](x) =12

(Ca Dγ

x[h](x) − Cx Dγ

b[h](x)), x ∈ (a, b)

where h′(x) denotes the first-order derivative of the function h with respect to x. In what follows,we consider only the Riesz-RL and the Riesz-Caputo fractional-order derivatives. Note that whenh is continuously differentiable and its first-order derivative is integrable, the RL and the Caputofractional-order derivatives are equivalent under the homogeneous boundary condition [29, 38].Due to the equivalency, both fractional-order derivatives are uniformly denoted by Dγ. More-over, since there is a singularity at the endpoint, we set h(a) = h(b) = 0 for the fractional-orderderivative.

In the discrete case, assume that u ∈ Rn×n under Dirichlet boundary condition. Accordingto [39,40], we give the discrete fractional-order derivatives at the point (i, j) with the order γ alongthe horizontal and the vertical direction, respectively

Dγhui, j =

12

i+1∑k=0

cγk ui−k+1, j +

n−i+2∑k=0

cγk ui+k−1, j

, i, j = 1, · · · , n

Dγv ui, j =

12

j+1∑k=0

cγk ui, j−k+1 +

n− j+2∑k=0

cγk ui, j+k−1

, i, j = 1, · · · , n

where cγk = (−1)k(γ

k

)for k = 0, 1, · · · , n + 1. Actually, we compute the coefficients cγk by the

following recursive relation

cγ0 = 1, cγk =

(1 −

1 + γ

k

)cγk−1, k = 1, 2, · · · .

Let c = cγ0 + cγ2, the discrete fractional-order derivative of u is formulated as

Dγ : Rn×n → R2n×n, Dγu :=(MuuM

)where

M =

2cγ1 c cγ3 · · · cγn

c 2cγ1. . .

. . ....

cγ3. . .

. . .. . . cγ3

.... . .

. . . 2cγ1 ccγn · · · cγ3 c 2cγ1

∈ Rn×n.

4

Due to the symmetry of the matrix M and the definition of the inner production on the matrix, i.e.,〈A, B〉 = tr(A>B) = (vecA)>vecB for all A, B ∈ Rn×n where vecA and vecB are vectors by stackingall the columns of the matrix A and B respectively, we have the following equations

〈Mu, v1〉 = tr(u>(Mv1)) = 〈u,Mv1〉

〈uM, v2〉 = (vec(uM>))>vecv2 = (vecu)>vec(v2M) = 〈u, v2M〉,

for any v1, v2 ∈ Rn×n. Finally, we obtain that

〈Dγu, v〉 = 〈Mu, v1〉 + 〈uM, v2〉 = 〈u,Mv1 + v2M〉, for all v =

(v1

v2

)∈ R2n×n.

Here we define the transform of the discrete fraction-order operator as (Dγ)>v = Mv1 + v2M.

2.2 Classic ADMM

According to [32–34, 41], the ADMM technique is widely applied for dealing with the followinggeneral constrained optimization problem

minx∈X,y∈Y

f (x) + g(y) (4)

s.t. Ax + By = b

where f (x) and g(y) are closed convex and lower semi-continuous functions, X ⊂ Rm and Y ⊂ Rn

are closed convex set, A ∈ Rl×m and B ∈ Rl×n are linear operators. By introducing a multiplierp ∈ Rl with the linear constraint Ax + By = b, the corresponding augmented Lagrangian functionis given by

L(x, y; λ) = f (x) + g(y) + p>(Ax + By − b) +δ

2‖Ax + By − b‖22 (5)

where β is a positive penalty parameter. Since the x-subproblem and the y-subproblem are decou-pled, the solution (xk+1, yk+1) of (4) is iteratively obtained by

xk+1 = arg minx

f (x) +δ

2

∥∥∥∥∥∥Ax + Byk − b +pk

δ

∥∥∥∥∥∥2

2,

yk+1 = arg miny

g(y) +δ

2

∥∥∥∥∥∥Axk+1 + By − b +pk

δ

∥∥∥∥∥∥2

2,

pk+1 = pk + τδ(Axk+1 + Byk+1 − b)

(6)

where τ > 0 controls the convergence speed. According to [42], if τ ∈(0,√

5+12

), the classic

ADMM algorithm is convergent.

3 The Proposed Models

In this paper, we build the target fraction-order gradient feature by integrating several noisy sourceimages. For obtaining the fused image, we assume that the fractional-order gradient informationof the fused image matches with the target fractional-order gradient feature, and the fused imageis closed to the predefined image u0. Moreover, the TV regularization preserves the image edges

5

while removes the noise [43]. Inspired by the works [23, 29–31], by combining with the TV reg-ularization, we propose a new variational model for simultaneously fusing and denoising (calledFTVL1 for short),

minu

∫Ω

|∇u| dx + α

∫Ω

|∇γu − w| dx +β

2

∫Ω

|u − u0|2 dx, (7)

where α, β are positive regularization parameters which control the balance between image fusionand denoising, w denotes the target fractional-order feature, u0 represents the preprocessed image.Similarly, we also present a variational model by using the difference between of ∇γu and w in thesense of the L2 norm (just referred to as FTVL2)

minu

E(u) =

∫Ω

|∇u| dx +α

2

∫Ω

|∇γu − w|2 dx +β

2

∫Ω

|u − u0|2 dx, (8)

Here, the first term∫Ω|∇u| dx in the models (7) and (8) represents the total variation (TV) regular-

ization, defined as follows∫Ω

|∇u| dx := sup ∫

Ω

u divϕ dx∣∣∣ϕ ∈ (C∞0 (Ω))2, ‖ϕ‖∞ ≤ 1

where (C∞0 (Ω))2 is the space of vector-valued functions with the compacted support in Ω. It isused for removing the Gaussian white noise in the source images.

The second terms∫Ω|∇γu−w| dx in (7) and

∫Ω|∇γu−w|2 dx in (8) are used to integrate all the

important salient features from the noisy source images. By combining the selection criterion [23],we construct the target fractional-order feature w from the noisy source images. Specifically,

assume that two source images f1 and f2, we compute the fractional-order gradient ∇γ fi =

(M fifiM

),

for i = 1, 2. According to the selection criterion [44, 45], we propose a relatively simple methodfor selecting the salient features from the noisy source images. The target fractional-order featurew is formulated as follows,

w = s∇γ f1 + (1 − s)∇γ f2

where s denotes a binary mask for fusing the salient features from the noisy source images. Bysmoothing and thresholding, the binary mask is given as

s(x) =

1, if K ∗ s > 0.5

0, otherwise,with s(x) =

1, if K ∗ (∇γ f1)2 > K ∗ (∇γ f2)2

0, otherwise,(9)

where ∗ represents the convolution and K is a mean kernel function, i.e., if y ∈ Ωx, then K(x, y) =

1/|Ωx|; otherwise K(x, y) = 0. The domain Ωx is a bounded neighborhood of x with area |Ωx|. Notethat when the number of the source images is more than two, we utilize the recursive method andonly incorporate two source images at one time. Furthermore, we choose the L2 norm to measurethe gap ∇γu − w in (8). This is because the L2 norm comes from Gaussian distribution by taking− log operaton. Specially, we consider the image “Cameraman” as the test image. The noisysource images are obtained by blurred the left part and right part with the out-of-focus kernel, thenadded the Gaussian white noise with mean zero and standard variance 20, 30, respectively. Thedesired fused image u is equal to the truth image, the target fractional-order feature w is obtainedby the above feature selection criterion. Figure 1 shows that the gap ∇γu − w in the horizontaldirection for the Gaussian white noise with standard variance 20 and 30. Their correspondinghistograms are shown in Figure 1(b) and (d) together with a fitting array containing random values

6

-150 -100 -50 0 50 100Histogram

0

500

1000

1500

2000

2500

3000∇

γu−w

Gaussian

(a)

-150 -100 -50 0 50 100Histogram

0

500

1000

1500

2000

2500

3000∇

γu−w

Gaussian

(b)

Figure 1. The corresponding histograms of the gaps ∇γu − w for the Gaussian white noise with standardvariance 20 (in the left side) and 30 (in the right side), respectively.

follows Gaussian distribution. We find that the histogram of ∇γu−w is matched with the histogramof an array obeyed Gaussian distribution. Therefore, we prefer to apply the L2 norm to measurethe gap ∇γu − w.

Furthermore, in order to obtain a better fused image, we assume that the fused image shouldbe close to a preprocessed image u0. Thus, a data-fitting term

∫Ω|u − u0|

2 dx is added to theproposed variational model. Since the preprocessed image u0 contains all the information fromthe noisy source images, the proposed model improves effectively the quality of the fused images.Furthermore, for preserving the image contrast, we combine the average contrast (AC) [23] withthe selection criterion. Specifically, if AC( f1) − AC( f2) > 0.5, then u0 = s f1 + (1 − s) f2 where ssatisfies the equation (9); otherwise, u0 is equal to the average of the noisy source images.

In the following, we give a theorem to illustrate the existence and uniqueness of the solutionof the proposed model (8). First, according to [28], we give some notations and definitions ofthe space of functions with γ-bounded variation on Ω. Let C`0(Ω,R2) denote the space of `-ordercontinuously differentiable functions with compact support in Ω ⊂ R2, K is the space of specialtest functions, which is formulated as

K :=ϕ ∈ C`0(Ω,R2)

∣∣∣ |ϕ| 6 1 for all x ∈ Ω

where |ϕ| =√ϕ2

1 + ϕ22. Then, the fractional-order total variation of u is defined by∫

Ω

|∇γu| dx := sup∫

Ω

−u divγϕ dx∣∣∣ϕ ∈ K

where divγϕ =∂γϕ1∂x +

∂γϕ2∂y , ∂γϕ1

∂x and ∂γϕ2∂y denote the fractional-order derivatives along the x-

direction and y-direction, respectively. Moreover, the space of functions with γ-bounded variationon Ω is given as follows,

BVγ(Ω) :=

u ∈ L1(Ω)∣∣∣ ∫

Ω

|∇γu| dx < +∞

.

With the γ-BV norm ‖u‖BVγ = ‖u‖L1 +∫Ω|∇γu| dx, the space BVγ(Ω) is a Banach space.

Theorem 3.1. Assume that u0 ∈ BVγ(Ω)∩L2(Ω) and w has a finite vector-valued Radon measure,then the minimization problem (8) exists an unique minimizer u∗ ∈ BVγ(Ω) ∩ L2(Ω).

7

Proof. In order to prove the existence and uniqueness of the solution, we first let uk ⊂ BVγ(Ω)∩L2(Ω) be a minimizing sequence of (8). Since E(0) is finite, the minimum of the energy functionE(u) is also finite. In other words, there exists a positive constant C such that

E(uk) =

∫Ω

|∇uk|dx +α

2

∫Ω

|∇γuk − w|2dx +β

2

∫Ω

|uk − u0|2dx 6 C.

Since w has the finite vector-valued Radon measure, we have that∫Ω

|w| := sup∫

Ω

w · ϕ dx∣∣∣ϕ ∈ K

< +∞.

Due to the fact that∫Ω|∇γuk −w| dx := sup

∫Ω

(−uk divγϕ + w · ϕ

)dx

∣∣∣ϕ ∈ K, then we obtain the

following inequalities as follows,∫Ω

|∇γuk| dx 6∫

Ω

|∇γuk − w| dx +

∫Ω

|w| 6 C (10)

Since∫Ω|∇uk| dx := sup

∫Ω

uk divϕ dx∣∣∣ϕ ∈ (C∞0 (Ω))2, ‖ϕ‖∞ ≤ 1

, we have∫

Ω

|uk − u0|2 dx 6 C. (11)

According to the triangle inequality and L2(Ω) ⊂ L1(Ω), we deduce that∫Ω

|uk|2 dx 6 C,∫

Ω

|uk| dx 6 C. (12)

Combining with (10), (11) and (12), we have that uk is bounded in BVγ(Ω)∩L2(Ω). Based on theweak∗ topology of BVγ(Ω) [28] and the reflexivity of L2(Ω), there exists a subsequence convergedto u∗ ∈ BVγ(Ω) ∩ L2(Ω) such that

uk BVγ−w∗−−−−−−→ u and uk L2(Ω)

−−−−→ u.

According to the lower semi-continuity of bounded variation, γ-bounded variation and L2-norm,we get ∫

Ω

|∇u∗| dx 6 lim infk→∞

∫Ω

|∇uk| dx,∫Ω

|∇γu∗ − w|2 dx 6 lim infk→∞

∫Ω

|∇γuk − w|2 dx,∫Ω

|u∗ − u0|2 dx 6 lim inf

k→∞

∫Ω

|uk − u0|2 dx.

Therefore, by Fatou’s lemma, we can conclude that E(u∗) 6 lim infk→∞ E(uk) and u∗ is a mini-mizer of (8). Moreover, since the energy function E(u) is obviously strictly convex, the minimizeru∗ is unique.

8

4 Proposed Algorithms

In this section, we develop an effective numerical algorithm for solving the proposed model (8)based on the classic ADMM algorithm. In fact, some other numerical methods can also be usedfor addressing the proposed model, such as the gradient descent method [43], the primal-dualmethod [46, 47], the split Bregman method [25] and the alternating minimization method [48,49]. Hereafter, we transform the proposed model into a discrete minimization. For the sake ofsimplicity, we still use the same notations in the model (8). The discrete minimization is rewrittenas follows,

minu∈Rn×n

‖Du‖1 +α

2‖Dγu − w‖2F +

β

2‖u − u0‖

2F . (13)

where u denotes the n × n gray-scale image and ‖ · ‖F represents the Frobenius norm. The discreteTV regularization term ‖Du‖1 under the zero boundary condition is formulated by

D : Rn×n → R2n×n, Du :=(

NuuN>

)where

M =

1−1 1

. . .. . .

−1 1

∈ Rn×n.

In order to apply the ADMM algorithm, we introduce a new auxiliary variable v ∈ R2n×n andgive the following constraint convex optimization problem

minu∈Rn×n,v∈R2n×n

‖v‖1 +α

2‖Dγu − w‖2F +

β

2‖u − u0‖

2F , (14)

s.t. v = Du

Then, let p ∈ R2n×n be the Lagrangian multiplier for the linear constraint v = Du, we give thecorresponding augmented Lagrangian function,

L(u, v, p) = ‖v‖1 +α

2‖Dγu − w‖2F +

β

2‖u − u0‖

2F + 〈p,Du − v〉 +

δ

2‖Du − v‖2F

where δ > 0 is the penalty parameter. Therefore, under the framework of ADMM, the minimiza-tion is divided into two separated subproblems.

For solving the v-subproblem, we utilize the shrinkage formula [48]. The solution is computediteratively by

vk+1 = arg minv‖v‖1 +

δ

2

∥∥∥∥∥∥Duk − v +pk

δ

∥∥∥∥∥∥2

F

= shrink(Duk +pk

δ,

1δ

) (15)

where shrink(x, y) = max(‖x‖F − y, 0) · x‖x‖F

, obeying the convention 0 · 00 = 0.

With respect to the u-subproblem, the minimization is written as

uk+1 = arg minu

α

2‖Dγu − w‖2F +

β

2‖u − u0‖

2F + 〈pk,Du − vk+1〉 +

δ

2‖Du − vk+1‖2F .

9

By the first-order optimality condition [50], we obtain the simplified linear system as follows,(α(Dγ)>Dγ + δD>D + βI

)u = α(Dγ)>w + D>(δvk+1 − pk) + βu0.

By some simple operations, we obtain the following Sylvester-like equation(αM2 + δN>N + βI

)u + u(αM2 + δN>N) = α(Dγ)>w + D>(δvk+1 − pk) + βu0. (16)

Then, we apply the conjugate gradient (CG) method [51] to solve the equation. In conclusion, thewhole algorithm for fusing the noisy source images is given as follows.

Algorithm 1. ADMM algorithm for solving (13)

1: Initialize u0, v0 and p0; set the parameters α, β, γ, δ.2: For k = 1, 2, . . . ,N, compute iteratively uk+1, vk+1, and pk+1 by

1) Compute vk+1 by the shrinkage formula (15);2) uk+1 is obtained by the CG method for solving (16);3) Update the multipliers:

pk+1 = pk + τδ(Duk+1 − vk+1).

Finally, motivated by the works in [42, 52], we give a following theorem to show the conver-gence of Algorithm 1.

Theorem 4.1. For the fixed parameter δ > 0 and τ ∈ (0,√

5+12 ), the ADMM algorithm for solving

the FTVL2 model (14) is convergent.

Proof. In order to prove the convergence of Algorithm 1, we first transform the proposed modelinto the general constrained convex problem. Specially, we set two functions as follows,

f (u) =α

2‖Dγu − w‖2F +

β

2‖u − u0‖

2F , g(y) = ‖v‖1.

Then with the linear constrained condition Du − v = 0, we can deduce that for fixed δ > 0 andτ ∈ (0,

√5+12 ), Algorithm 1 is convergent.

Note that although the u-subproblem is approximatively solved by the conjugate gradient (CG)method, Algorithm 1 is empirically convergent.

In addition, for fair comparison with the FTVL1 model, we also apply the classic ADMMalgorithm to solve (7). The corresponding discrete minimization problem is given as follows:

minu∈Rn×n

‖Du‖1 + α‖Dγu − w‖1 +β

2‖u − u0‖

2F . (17)

By introducing two variables v and t, we transforms into the following constrained minimizationproblem

minu,v,t‖v‖1 + α‖t‖1 +

β

2‖u − u0‖

2F , (18)

s.t. v = Du, t = Dγu − w.

10

Therefore, the augmented Lagrangian function of (18) is formulated as

L(u, v, t, p1, p2) = ‖v‖1 + α‖t‖1 +β

2‖u − u0‖

2F + 〈p1,Du − v〉 +

δ1

2‖Du − v‖2F

+ 〈p2,Dγu − w − t〉 +δ2

2‖Dγu − w − t‖2F

where p1 and p2 denotes the multipliers, δ1 and δ2 denotes the positive penalty parameters. Simi-larly, we apply the following algorithm to solve the FTVL1 model.

Algorithm 2. ADMM algorithm for solving (17)

1: Initialize u0, v0, t0, p01 and p0

2; set the parameters α, β, γ, δ1, δ2.2: For k = 1, 2, . . . ,N, compute iteratively uk+1, vk+1, tk+1, pk+1

1 and pk+12 by

1) Compute vk+1 and tk+1 by the shrinkage formulas

vk+1 = shrink(Duk +pk

1

δ1,

1δ1

);

tk+1 = shrink(Dγuk − w +pk

2

δ2,α

δ2);

2) uk+1 is obtained by the CG method;3) Update the multipliers:

pk+11 = pk

1 + τδ1(Duk+1 − vk+1),

pk+12 = pk

2 + τδ2(Dγuk+1 − w − tk+1).

5 Experiments

In this section, we present several numerical experiments to illustrate the superior performance ofthe proposed method for fusing the noisy source images. All numerical experiments are performedunder Windows 8 and Matlab (R2015b) running on a desktop with 3.40GHz Intel Core i3-2130CPU and 4G RAM memory. In the experiments, all the pixel values of the test source imagesbelong to the interval [0, 255]. In order to show the efficiency of the proposed method, we comparewith some existing popular algorithm including the WFTV model [20] and the FSGF model [23].For the FSGF model, we first denoise the source images and then fuse the denoised images.

In real applications, it is not easy to quantitatively measure the quality of the fused results dueto the absence of the true original image. According to [53], for evaluating the quality of the fusedresults, we just consider two of all the objective fusion metrics. The mutual information basedfusion metric [54, 55] is defined by

QMI = MI(u∗, f1) + MI(u∗, f2),

where u∗ is the fused image, f1 and f2 are the reference images. The mutual information operationMI(x, y) can be expressed as

MI(x, y) = H(x) + H(y) − H(x, y)

11

with

H(x) = −∑

x

p(x) log2 p(x)

H(x, y) = −∑x,y

p(x, y) log2 p(x, y),

where p(x) is the marginal probability distribution function and p(x, y) is the joint probabilitydistribution function. The higher MI values implies the better fused results. Moreover, by com-bining with the principal (maximum and minimum) moments of the image phase congruency, anew fusion metric [56, 57] is defined as a product,

QP = (Pp)α(PM)β(Pm)γ,

where p is the phase congruency, M is the maximum moment, and m is the minimum moment,Pp = maxCp

f1u∗ ,Cpf2u∗ ,C

psu∗, PM = maxCM

f1u∗ ,CMf2u∗ ,C

Msu∗, Pm = maxCm

f1u∗ ,Cmf2u∗ ,C

msu∗. Note

that s is the maximum-select map. The parameters α, β and γ can be tuned according to theimportance of the three components. The correlation coefficient between two sets x and y is givenby

Ckxy =

σkxy + C

σkxσ

ky + C

,

where k ∈ p,M,m, σx and σy are their respective standard variances, σxy are the covariance of xand y, C > 0 is a constant.

For the fair compasions, we set the same maximum iteration number as 10 in the FSGF model,the FTVL1 model and the FTVL2 model. In Algorithm 1 and 2, since the parameters α and β

balance the trade-off between the TV regularization and the data-fitting terms, we manually tunethem for obtaining the good fused results. Then we set 1 ≤ γ ≤ 2 in the fractional-order gradientoperator and 0 ≤ δ ≤ 10−5. Furthermore, since the parameter τ controls the convergence speed,we set τ = 1.618 which the ADMM algorithm 1 converges faster than τ = 1. In addition, theiterative number of the conjugate gradient method is equal to 5 for solving the u-subproblem.

5.1 Experiment on the Image Cameraman

The test image “Cameraman” is blurred in the left part and right part respectively with a Gaussiankernel (mean zero, standard deviation σ = 2, kernel size 5 × 5). Then the Gaussian noise is addedto the blurring images, where the noise level σ ∈ 15, 20, 25, 30. We compare the fused resultsby using the FSGF, WFTV, FTVL1 and FTVL2 model. Table 1 lists the values of QMI and QP byusing four different methods for the test image “Cameraman”. By comparing the values of QMI

and QP, the FTVL1 and FTVL2 model obtain the higher results than the FSGF and WFTV model.Moreover, the FTVL2 model obviously improves the values of QMI .

Figure 2 shows the fused images by using different methods for the test image “Cameraman”,where the noise level σ = 30. We find that the noise is not thoroughly removed by the FSGFmodel, but the WFTV, FTVL1 and FTVL2 model are able to eliminate the white Gaussian noisein the source images (b) and (c). Unfortunately, the WFTV model suffers from some undesiredfalse edges. The FTVL1 model products some artifacts in the background. The fused image byusing the FTVL2 model is obviously better than the the WFTV and FTVL1 model. The FTVL2model not only efficiently removes the white Gaussian noise, but also fuses highly the importantfeatures from two source images.

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Table 1. The values of QMI and QP by using different methods for the test image “Cameraman” .

QMI QP

Noise FSGF WFTV FTVL1 FTVL2 FSGF WFTV FTVL1 FTVL2

15 5.4948 5.8548 5.9135 6.0294 0.5060 0.4955 0.5132 0.519520 5.1070 5.6269 5.6509 5.8047 0.4064 0.4353 0.4398 0.449025 4.7996 5.4410 5.4244 5.6272 0.3451 0.3935 0.3828 0.400730 4.5384 5.2760 5.2344 5.4845 0.2985 0.3583 0.3411 0.3672

(a) (b) (c)

(d) (e) (f) (g)

Figure 2. Comparison of different methods for fusing and denoising, the noise standard variance σ = 30.(a) The original image; (b) and (c) two source images by blurring the left part and right part; (d) FSGF [23];(e) WFTV [20]; (f) FTVL1; (g) FTVL2.

For illustrating the superior performance of the proposed model, we give the correspondingedge maps of the fused images by the canny edge detector in Figure 3. By the edge maps com-parison of the fused images, the FTVL2 model outperforms the three other methods. In the edgemap of the FSGF model, there still exists much noise. However, although the region of the grassis smoothed by the WFTV, FTVL1 and FTVL2 model, there are no much artifacts in the region ofthe sky and the black coat for the FTVL2 model.

5.2 Experiment on the Real Image

In this section, we do several experiments on the real images for simultaneously denoising andfusing. Figure 4 shows the test real images, including three pairs of multi-focus images andfive pairs of multi-modal images, which can be found in the website http://www.quxiaobo.org/index.html. The noisy images are obtained by adding the white Gaussian noise with thestandard variance σ ∈ 15, 20, 25, 30, respectively. First, we list Table 2 shown the values ofQMI and QP by using four different model for the real multi-focus images. Our proposed FTVL2model obtains the highest values of QMI by comparing with the FSGF, WFTV and FTVL1 model.

13

(a) (b)

(c) (d) (e)

Figure 3. Edge maps of the fused images in Figure 2 by using the canny edge detector. (a) The originalimage; (b) FSGF [23]; (c) WFTV [20]; (d) FTVL1; (f) FTVL2.

But, by comparing with the values of QP, we nearly get the higher values than the other models.Especially, when the noise level is fairly large, the results of the proposed model shows the notableimprovement.

Table 2. The values of QMI and QP by using different methods for the real multi-focus images.

QMI QP

Image Noise FSGF WFTV FTVL1 FTVL2 FSGF WFTV FTVL1 FTVL2

15 4.0347 3.9987 4.1202 4.1370 0.3192 0.3299 0.3637 0.3634Leaves 20 3.8820 3.8466 3.9819 4.0141 0.2784 0.2762 0.2956 0.2965

25 3.7433 3.7377 3.8746 3.9069 0.2382 0.2370 0.2430 0.247630 3.6106 3.6350 3.7671 3.8061 0.2075 0.2086 0.2032 0.2129

15 5.3383 6.0771 6.0383 6.0930 0.2698 0.3343 0.3228 0.3401Book 20 5.0130 5.8697 5.8207 5.9102 0.2072 0.2952 0.2868 0.2954

25 4.7648 5.6822 5.6342 5.7364 0.1678 0.2649 0.2516 0.265930 4.5413 5.5135 5.4604 5.5848 0.1401 0.2415 0.2265 0.2514

15 4.8575 5.6256 5.6138 5.6456 0.2875 0.3722 0.3503 0.3499Plane 20 4.4670 5.3921 5.3303 5.3998 0.2372 0.3335 0.3007 0.3107

25 4.1008 5.1903 5.0720 5.2062 0.1939 0.3022 0.2641 0.286430 3.8178 5.0092 4.8324 5.0186 0.1573 0.2754 0.2350 0.2519

Figure 5 and 7 show the fused images by applying four different models, where the whiteGaussian noise σ = 20, 30. Since the FSGF model is just used to fuse the noiseless source images,

14

(a) Leaves (204 × 204) (b) Book (256 × 256) (c) Plane (160 × 160) (d) Brain (256 × 256)

(e) Crossroad (256 × 256) (f) Hill (240 × 240) (g) Grass (256 × 256) (h) Plant (256 × 256)

Figure 4. Source images.

(a) (b) (c) (d) (e) (f)

Figure 5. Comparison of different models for the images “Leaves”, where the white Gaussian noise σ = 20(in the 1st row) and σ = 30 (in the 2nd row), respectively. (a) and (b): two input source images; (c)FSGF [23]; (d) WFTV [20]; (e) FTVL1; (f) FTVL2.

we find that the fused result (c) still retains some noise. The WFTV model is able to remove thenoise, but oversmoothes the details of images. Compared with the FSGF and WFTV model, thefused images of the FTVL1 and FTVL2 model are obviously improved. Furthermore, the fusedimages of the FTVL2 model preserve a little clearer texture than the FTVL1 model. In conclusion,the proposed FTVL2 model can effectively remove the white Gaussian noise as well as fuse thesalient feature from the source images. Especially, we give the zoomed parts of the fused imagesby using different models, shown in Figure 6. The texture of the leaves by the FTVL2 model isclearer than the FTVL1 model.

Secondly, we do the experiments for the test multi-modal images by using four different mod-els. Table 3 shows the values of QMI and QP by applying four different models, where the noiselevel σ = 15, 20, 25, 30. The FTVL1 and FTVL2 model almost obtain the higher values with re-spect to QMI and QP than the FSGF and WFTV model. However, the fused results of the proposedFTVL2 model is better than the FTVL1 model. Due to this reason, we observe that the FTVL2model is more suitable for fusing the noisy source images.

Figure 8 shows the Comparison of different models for fusing the noisy computed tomography(CT) image and the noisy magnetic resonance imaging (MRI) image, where the white Gaussiannoise σ = 20 and σ = 30, respectively. Figure 8(a) shows the CT images which well provides thedetail structure of the body’s bone, Figure 8(b) shows the MRI images which mainly illuminatesthe detail structure of the body’s soft tissue. Figure 8(c-f) show the fused results by applying the

15

(a) (b) (c) (d)

Figure 6. Comparison of zoomed fused images in Figure 5, where the white Gaussian noise σ = 20 (in the1st row) and σ = 30 (in the 2nd row), respectively. (a) FSGF [23]; (b) WFTV [20]; (c) FTVL1; (d) FTVL2.

FSGF model [23],the WFTV model [20], the FTVL1 model and the FTVL2 model. Figure 8(c)is the fused result of the FSGF model, there still retains some noise. But from Figure 8(d-f), thenoise is effectively removed by the WFTV model [20], the FTVL1 model and the FTVL2 model.Unfortunately, the WFTV model darken the fused image and reduce the image contrast. TheFTVL1 model and the FTVL2 model gets the better fused images. But the FTVL1 model productsthe undesired artifacts in the fused images. The FTVL2 model fuses the abundant information andsalient feature of the noisy source images, especially by comparing with the zoomed parts of thefused image in Figure 9. By compared with the FTVL1 model, the fused images of the FTVL2model better fit with the quality perception of the human visual system (HVS). Therefore, theproposed model effectively removes the noise as well as fuses the salient detail of the CT imagesand MRI images.

(a) (b) (c) (d) (e) (f)

Figure 7. Comparison of different models for the images “Book”, where the white Gaussian noise σ = 20(in the 1st row) and σ = 30 (in the 2nd row), respectively. (a) and (b): two input source images; (c)FSGF [23]; (d) WFTV [20]; (e) FTVL1; (f) FTVL2.

16

Table 3. The values of QMI and QP by using different models for the real multi-modal images.

QMI QP

Image Noise FSGF WFTV FTVL1 FTVL2 FSGF WFTV FTVL1 FTVL2

15 2.6017 2.5786 2.7504 2.7885 0.2057 0.2014 0.2192 0.2133Brain 20 2.4345 2.4550 2.6302 2.6861 0.1553 0.1698 0.1788 0.1735

25 2.3198 2.3498 2.5310 2.6105 0.1268 0.1481 0.1496 0.152930 2.2203 2.2695 2.4614 2.5499 0.1035 0.1314 0.1285 0.1385

15 3.1104 2.6442 3.1401 3.1529 0.4861 0.3601 0.4891 0.4943Grass 20 2.9784 2.5556 2.9814 2.9981 0.4227 0.2943 0.4123 0.4208

25 2.8489 2.4823 2.8461 2.8892 0.3602 0.2449 0.3448 0.358630 2.7402 2.4173 2.7420 2.7936 0.3079 0.2094 0.2894 0.3093

15 2.7077 2.1029 2.8007 2.8271 0.2774 0.2531 0.2958 0.3012Crossroad 20 2.4283 1.9897 2.5829 2.6417 0.2132 0.2185 0.2481 0.2492

25 2.2137 1.9043 2.4015 2.4753 0.1669 0.1916 0.2124 0.214730 2.0501 1.8284 2.2745 2.3578 0.1388 0.1709 0.1813 0.1901

15 2.5245 1.7110 2.5897 2.9196 0.1073 0.1091 0.1105 0.01147Hill 20 2.3027 1.6892 2.4380 2.4840 0.0797 0.0925 0.0901 0.0926

25 2.1320 1.6677 2.3204 2.3704 0.0611 0.0806 0.0748 0.080630 2.0114 1.6477 2.2236 2.2957 0.0516 0.0724 0.0656 0.0737

15 3.1427 3.1157 3.3199 3.3327 0.3800 0.3734 0.3935 0.4000Plant 20 3.1056 2.9502 3.1316 3.1385 0.3175 0.3122 0.3117 0.3194

25 2.9366 2.8258 2.9659 2.9919 0.2527 0.2553 0.2531 0.263930 2.8034 2.7230 2.8278 2.8532 0.2137 0.2197 0.2083 0.2208

Figure 10 and 11 are the Comparison of different models for fusing the noisy visible imageand the noisy infrared image. The visible image captures many spatial details and backgroundinformation, while the infrared image easily determines the position of thermal objects. The visibleimage is different from the infrared image. And the information of the visible image and theinfrared image is complementary. Obviously, for the FSGF model, the noise is not thoroughlyremoved. The WFTV model reduce the contrast of the fused image. But, for the proposed FTVL1and FTVL2 models, all the salient features of are fairly clear. Furthermore, the proposed FTVL2model obtains the better fused image without the undesired artifacts.

6 Conclusion

In this paper, according to the underlying properties of the fractional-order derivative, we as-sume that the synthetical fractional-order gradient information from the source images fits withthe fractional-order gradient information of the fused image in the sense of L2 norm. By combin-ing the total variation (TV) regularization for removing the noise with the fidelity term for makingthe fused image close a predefined image, we present a novel convex variational model for fus-ing the noisy source images. Then, an alternating direction method of multiplier (ADMM) isdeveloped for solving the proposed FTVL2 model. Numerical experiments show that our methodoutperform the state-of-the-art variational methods in terms of visual and quantitative measures.

Acknowlegenments

We would like to thank Fang Li and Weiwei Wang for providing the software codes [23] and [20].

17

(a) (b) (c) (d) (e) (f)

Figure 8. Comparison of different models for fusing and denoising in the images “Brain”, where whiteGaussian noise σ = 20 (in the 1st row) and σ = 30 (in the 2nd row), respectively. (a) and (b): two sourceimages; (c) FSGF [23]; (d) WFTV [20]; (e) FTVL1; (f) FTVL2.

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